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We are asked to find a polynomial \( P(x) \) of degree 5 with the following conditions:

- It has leading coefficient 1.
- It has roots of multiplicity 2 at \( x = 1 \) and \( x = 0 \).
- It has a root of multiplicity 1 at \( x = -3 \).

### Step-by-step approach:

1. **Roots with Multiplicities:**
   - The root at \( x = 1 \) has multiplicity 2, so the factor is \( (x - 1)^2 \).
   - The root at \( x = 0 \) has multiplicity 2, so the factor is \( x^2 \).
   - The root at \( x = -3 \) has multiplicity 1, so the factor is \( (x + 3) \).

2. **Forming the Polynomial:**
   Since the leading coefficient is 1, the polynomial can be written as:
   \[
   P(x) = (x - 1)^2 \cdot x^2 \cdot (x + 3)
   \]

3. **Expanding the Polynomial:**
   - First, expand \( (x - 1)^2 = x^2 - 2x + 1 \).
   - Multiply \( (x^2 - 2x + 1) \) by \( x^2 \):
     \[
     (x^2 - 2x + 1) \cdot x^2 = x^4 - 2x^3 + x^2
     \]
   - Now multiply the result by \( (x + 3) \):
     \[
     (x^4 - 2x^3 + x^2) \cdot (x + 3) = x^5 + 3x^4 - 2x^4 - 6x^3 + x^3 + 3x^2
     \]
     Simplifying this:
     \[
     P(x) = x^5 + x^4 - 5x^3 + 3x^2
     \]

Thus, a possible formula for \( P(x) \) is:
\[
P(x) = x^5 + x^4 - 5x^3 + 3x^2
\]

Would you like more details on any of the steps? Here are some related questions:

1. What does it mean for a root to have a certain multiplicity?
2. How do you determine the degree of a polynomial from its factors?
3. What are the characteristics of a polynomial with a given leading coefficient?
4. How do you expand products of polynomials step-by-step?
5. What would happen if the leading coefficient were not 1?

**Tip:** When expanding polynomials, always group terms of the same degree to simplify easily!

The polynomial of degree 5, P(x), has leading coefficient 1, has roots of multiplicity 2 at x = 1 and x = 0, and a root of multiplicity 1 at x = -3. Find a possible formula for P(x).

Learn how to find a possible formula for a polynomial of degree 5 with roots of multiplicity 2 at x = 1 and x = 0, and a root of multiplicity 1 at x = -3. This problem demonstrates polynomial expansion and root multiplicity, suitable for Grades 10-12.

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